Problem: Find $\int \dfrac{1}{\sqrt{-x^2-4x+21}}\,dx$. Choose 1 answer: Choose 1 answer: (Choice A) A $\dfrac15\text{arctan}\left(\dfrac{x+2}{5}\right)+C$ (Choice B) B $\text{arctan}\left(\dfrac{x+2}{5}\right)+C$ (Choice C) C $\text{arcsin}\left(\dfrac{x+2}{5}\right)+C$ (Choice D) D $\dfrac15\text{arcsin}\left(\dfrac{x+2}{5}\right)+C$
The integrand is in the form $\dfrac{1}{\sqrt{p(x)}}$ where $p(x)$ is a quadratic expression. This suggests that we should rewrite $p(x)$ by completing the square. Specifically, we will rewrite $p(x)$ as $ k^2-(x+ h)^2$. Then, we will be able to integrate using our knowledge of the derivative of the inverse sine function: $\int \dfrac{1}{\sqrt{ k^2-x^2}}\,dx=\text{arcsin}\left(\dfrac{x}{ k}\right)+C$ [Why is this formula true?] By setting $u=x+ h$ and using $u$ -substitution, we get the following formula: $\int \dfrac{1}{\sqrt{ k^2-(x+ h)^2}}\,dx=\text{arcsin}\left(\dfrac{x+ h}{ k}\right)+C$ We start by rewriting $p(x)$ as $ k^2-(x+ h)^2$ : $\begin{aligned} -x^2-4x+21&=21-(x^2+4x) \\\\ &=21+4-(x^2+4x+4) \\\\ &=25-(x+2)^2 \\\\ &={5}^2-(x+{2})^2 \end{aligned}$ Now we can find the integral: $\begin{aligned} &\phantom{=}\int \dfrac{1}{\sqrt{-x^2-4x+21}}\,dx \\\\ &=\int\dfrac{1}{\sqrt{{5}^2-(x+{2})^2}}\,dx \\\\ &=\text{arcsin}\left(\dfrac{x+{2}}{{5}}\right)+C \end{aligned}$ In conclusion, $\int \dfrac{1}{\sqrt{-x^2-4x+21}}\,dx=\text{arcsin}\left(\dfrac{x+2}{5}\right)+C$